Electrostatics

Step 1: Understanding the Concept:
Surface charge density ( ) is a measure of the amount of electric charge distributed over a given surface area. It is defined as the charge per unit area.
Step 2: Key Formula or Approach:
The formula for surface charge density is:
where:
- is the surface charge density.
- is the total charge on the surface.
- is the total surface area.
For a sphere, the surface area is given by , where is the radius.
Step 3: Detailed Explanation:
First, we need to list the given values and convert them to S.I. units.
- Charge, .
- Radius, .
We can notice that . This approximation can simplify the calculation. So let's use C.
Next, calculate the surface area ( ) of the sphere:
Now, calculate the surface charge density ( ):
The terms cancel out.
Step 4: Final Answer:
The surface density of charge on the sphere is C/m .

Step 1: Understanding the Concept:
The capacity, or capacitance ( ), of a condenser (an older term for capacitor) is a measure of its ability to store electric charge. It is defined as the ratio of the magnitude of the charge ( ) on either conductor to the magnitude of the potential difference ( ) between them: . However, the capacitance itself does not depend on or ; it is determined by the physical characteristics of the capacitor.
Step 2: Key Formula or Approach:
To identify the factors, we can look at the formula for the capacitance of a standard parallel-plate capacitor, which illustrates the dependencies clearly:
where:
- is the capacitance.
- (or ) is the dielectric constant of the material between the plates.
- is the permittivity of free space (a constant).
- is the permittivity of the dielectric medium.
- is the area of overlap of the plates.
- is the distance between the plates.
Step 3: Detailed Explanation of Factors:
From the formula, we can identify the key factors:
1. Geometrical Factors: The capacitance depends on the size, shape, and relative positioning of the conductors. For the parallel-plate capacitor, these are:
- Area of the plates ( ): The capacitance is directly proportional to the area of the plates ( ). A larger area allows more charge to be stored for the same potential difference.
- Distance between the plates ( ): The capacitance is inversely proportional to the distance between the plates ( ). Bringing the plates closer increases the electric field between them, which allows more charge to be stored for a given voltage.
2. Dielectric Medium:
- Permittivity of the medium ( ): The capacitance is directly proportional to the permittivity of the dielectric material placed between the conductors ( ). A dielectric material with a higher dielectric constant ( ) increases the capacitance because it reduces the effective electric field, allowing more charge to be stored at the same potential difference.
Step 4: Final Answer:
Therefore, the two main factors determining the capacitance of a condenser are its physical geometry (area of plates, distance between them) and the nature of the dielectric medium separating them.

Part 1: Capacity of a Conductor
Step 1: Understanding the Concept:
The capacity, or capacitance, of a conductor is its intrinsic ability to store electric charge. When charge is given to an isolated conductor, its electric potential increases.
It is observed that the potential ( ) of the conductor is directly proportional to the charge ( ) given to it.
Step 2: Definition:
The constant of proportionality, , is called the capacitance of the conductor. It is defined as the ratio of the charge given to the conductor to the potential raised on it.
Thus, the capacity of a conductor is numerically equal to the amount of charge required to raise its potential by one unit. Its S.I. unit is the Farad (F).
Part 2: Expression for the Capacity of a Cylindrical Condenser
Step 1: Setup and Assumptions:
A cylindrical condenser (capacitor) consists of two coaxial hollow conducting cylinders. Let the inner cylinder have radius 'a' and the outer cylinder have radius 'b'. Let the length of the cylinders be 'L', and assume so that edge effects can be neglected. The space between the cylinders is filled with a dielectric medium of permittivity . Let a charge of be given to the inner cylinder and to the outer cylinder.

Step 2: Calculating Electric Field:
To find the electric field at a point at a radial distance ( ) from the axis, we consider a cylindrical Gaussian surface of radius and length .
The linear charge density is .
According to Gauss's law:
The direction of the electric field is radially outwards.
Step 3: Calculating Potential Difference:
The potential difference ( ) between the inner and outer cylinders is the work done in moving a unit positive charge from the outer cylinder to the inner cylinder.
Step 4: Finding the Capacitance:
Using the definition of capacitance, :
If the medium is a vacuum or air, , and the expression becomes:
This is the required expression for the capacity of a cylindrical condenser.

Step 1: Understanding the Concept:
A Van de Graaff generator is a device designed to create a very large static electric potential (voltage). It operates by using a moving belt to transport electric charge from a source to a large, hollow metal sphere.
Step 2: Detailed Explanation:
Voltage Production: The generator accumulates a large amount of charge on the outer surface of its spherical terminal. The potential of a sphere is given by . By continuously transporting charge ( ) to the sphere, a very high potential difference (voltage), often in the range of millions of volts, is built up with respect to the ground. Its primary function is to produce this high voltage.
Current Production: The charge is transported mechanically by a physical belt. The rate at which charge can be moved is limited, so the resulting electric current ( ) is very small, typically in the microampere ( ) range.
Conclusion: The defining characteristic of a Van de Graaff generator is its ability to produce extremely high voltages. The current it can deliver is, by contrast, very low. Therefore, among the given choices, "Only high voltage" is the best description of its output.
Step 3: Final Answer:
A Van de Graaff generator produces a very high voltage but a very low current. Option (B) correctly identifies its main output.

Step 1: Understanding the Concept:
Permittivity ( ) is a measure of how an electric field affects, and is affected by, a dielectric medium. The question asks for the SI unit of permittivity, often referring to the permittivity of free space, .
Step 2: Key Formula or Approach:
We can derive the unit of permittivity from Coulomb's Law, which describes the electrostatic force ( ) between two point charges ( and ) separated by a distance ( ).
Step 3: Detailed Explanation:
First, we rearrange Coulomb's Law to solve for :
Now, we substitute the SI units for each physical quantity into the equation. The term is a dimensionless constant.
\begin{itemize} \item Unit of Force ( ): Newton (N) \item Unit of Charge ( ): Coulomb (C) \item Unit of Distance ( ): meter (m) \end{itemize} Substituting these units:
This unit can be written in exponent notation as:
Step 4: Final Answer:
The SI unit of permittivity is C N m . This matches option (D).

Step 1: Understanding the Concept:
Electric potential ( ) at a point in an electric field is defined as the work done in bringing a unit positive charge from infinity to that point. We need to find how this potential varies with distance from a single point charge.
Step 2: Key Formula or Approach:
The electric potential at a distance from a source point charge is given by the formula: where is Coulomb's constant ( ). So, .
Step 3: Detailed Explanation:
From the formula , we can see that for a given charge , the potential is inversely proportional to the distance . This means that as the distance from the charge increases, the electric potential decreases.
Option (D), inversely proportional to , describes the electric field strength ( ), not the potential.
Step 4: Final Answer:
The electric potential due to a point charge is inversely proportional to the distance from the charge. Therefore, option (B) is correct.

Step 1: Understanding the Concept:
When small charged conducting drops combine to form a larger drop, the total volume is conserved. The capacitance of a spherical conductor is proportional to its radius. We can find the radius of the big drop in terms of the small drops' radius and then find its capacitance.
Step 2: Key Formula or Approach:
Let be the radius and be the capacity of each small drop.
Let be the radius and be the capacity of the big drop.
The capacity of a spherical conductor is given by .
The volume of small drops equals the volume of the big drop: This simplifies to .
Step 3: Detailed Explanation:
Given:
Number of drops, .
Capacity of each small drop, .
First, find the radius of the big drop ( ) in terms of the small drop radius ( ): The radius of the big drop is 4 times the radius of a small drop.
Now, find the capacity of the big drop ( ): Substitute : Since , we have: Finally, substitute the given value of : Step 4: Final Answer:
The capacity of the big drop will be 20 µF. Therefore, option (D) is correct.

Step 1: Understanding the Concept:
Electrostatic potential energy ( ) of a system of two point charges is the work done in assembling the charges from an infinite separation to their current positions.
Step 2: Key Formula or Approach:
The formula for the electrostatic potential energy between two point charges and separated by a distance is:
where is Coulomb's constant.
Step 3: Detailed Explanation:
The change in potential energy when the distance is increased depends on the product of the charges , which is determined by whether the charges are like (both positive or both negative) or unlike (one positive, one negative).
Case 1: Like Charges
If both charges are positive or both are negative, their product is positive.
In this case, is positive. As the distance increases, the value of the fraction decreases. Consequently, the potential energy decreases. This makes sense, as the repulsive force does positive work when the charges move apart.
Case 2: Unlike Charges
If one charge is positive and the other is negative, their product is negative.
In this case, is negative. As the distance increases, the value of decreases, making less negative (i.e., it moves closer to zero). A change from a more negative value to a less negative value is an increase. This makes sense, as work must be done against the attractive force to separate the charges.
Step 4: Final Answer:
Since the question does not specify whether the charges are like or unlike, the potential energy may either increase (for unlike charges) or decrease (for like charges) when the distance is increased. Thus, option (C) is correct.

Step 1: Understanding the Concept:
To obtain the maximum equivalent capacitance from a set of capacitors, they should be connected in parallel. To obtain the minimum equivalent capacitance, they should be connected in series.
Step 2: Key Formula or Approach:
Given three capacitors, each with capacitance .
For Parallel Combination (Maximum Capacitance): The equivalent capacitance is the sum of individual capacitances. For Series Combination (Minimum Capacitance): The reciprocal of the equivalent capacitance is the sum of the reciprocals of individual capacitances. Step 3: Detailed Explanation:
Calculation for Maximum Capacitance (Parallel):
Connect all three 6 µF capacitors in parallel. Calculation for Minimum Capacitance (Series):
Connect all three 6 µF capacitors in series. To find , we take the reciprocal: Step 4: Final Answer:
The minimum capacitance obtained is 2 µF, and the maximum capacitance obtained is 18 µF. Therefore, option (C) is the correct choice.

Step 1: Understanding the Concept:
This question requires the application of Gauss's Law in electrostatics. Gauss's Law relates the total electric flux ( ) through any closed surface (called a Gaussian surface) to the net electric charge ( ) enclosed by that surface.
Step 2: Key Formula or Approach:
Gauss's Law is stated mathematically as:
Where:
is the total electric flux.
is the total charge enclosed within the surface.
is the permittivity of the medium. For air or vacuum, we use the permittivity of free space, .
Step 3: Detailed Explanation:
According to the problem statement:
- The charge is a "unit positive charge", which means C.
- The medium is "air", so we use the permittivity .
Substituting these values into Gauss's Law:
This can be written using a negative exponent as:
Step 4: Final Answer:
The total electric flux is . This corresponds to option (B).

Step 1: Understanding the Concept:
This question asks for the definition of a fundamental quantity in electrostatics. We need to identify the physical quantity that is defined as the electrostatic force experienced by a charge divided by the magnitude of that charge.
Step 2: Detailed Explanation:
Let's analyze the definitions of the given options:
- Electric current is the rate of flow of electric charge ( ). It is not force per unit charge.
- Electric potential at a point is the work done in moving a unit positive charge from a reference point (usually infinity) to that point ( ). It is energy per unit charge, not force per unit charge.
- Electric field (or electric field intensity, ) at a point is defined as the electrostatic force ( ) experienced by a small positive test charge ( ) placed at that point, divided by the magnitude of the test charge.
This exactly matches the description "force, acting on per unit charge".
- Electric space is not a standard term in physics for a physical quantity.
Step 3: Final Answer:
The force per unit charge is the definition of the electric field. Therefore, option (C) is correct.

Electrons are negatively charged particles, and according to Coulomb’s law, like charges repel each other with a force that is inversely proportional to the square of the distance between them. This fundamental electrostatic repulsion causes electrons within a beam to push away from one another. When multiple electron beams are moving parallel to each other with uniform velocity, each electron in one beam experiences repulsive forces from the electrons in the neighboring beams. These forces act perpendicular to the direction of motion and cause the beams to exert a lateral push against each other. Because the electrons are all moving in the same direction with approximately the same velocity, the mutual repulsion is not balanced by any attractive forces. This leads to a gradual spreading or divergence of the electron beams as they travel. In practical applications, such as cathode ray tubes or electron microscopes, this effect limits the ability to keep electron beams narrowly focused over long distances. Engineers often use magnetic or electrostatic lenses to counteract this divergence and maintain beam coherence. In summary, the tendency of electron beams to diverge when moving parallel with uniform velocity arises from the mutual electrostatic repulsion between electrons carrying like negative charges, which pushes the beams apart laterally as they propagate.

A picofarad (pF) is a unit of capacitance commonly used to measure very small capacitances. It is defined as: Here, the farad (F) is the SI unit of capacitance, and the prefix "pico-" denotes a factor of . Thus, one picofarad is one trillionth of a farad.

The capacitance of a condenser depends on the shape, size, and distance between the plates, but it does not depend on the charge present on the plates. Capacitance is a geometric property defined by the formula: where:

is the capacitance,
is the permittivity of free space,
is the area of the plates,
is the distance between the plates.
This formula shows that capacitance increases with larger plate area and decreases with greater separation between the plates.

The capacitance of a spherical conductor is given by: where is the radius. Given: Solving for : Since 0.9 cm is close to 1 cm, the closest given option is (D) 11 cm (probably a typo in options; correct value is about 0.9 cm). Please verify options.

Step 1: The charge on one electron is . This is a fundamental physical constant representing the elementary charge carried by a single electron, and the negative sign indicates that the electron is negatively charged.
Step 2: When electrons are added to an initially uncharged body, each carrying a charge of , the total charge acquired by the body is calculated as: Here, is the number of electrons added, and is the magnitude of the charge on one electron. The product gives the total charge , and the negative sign indicates that the charge is negative.
Step 3: Therefore, the body gains a negative charge of 1.6 C. This means that the body is now negatively charged due to the excess of electrons.

The intensity of the electric field at any point is defined as the force experienced by a unit positive test charge placed at that point. It is a vector quantity that indicates both the magnitude and direction of the force exerted on the charge. Mathematically, the electric field intensity is given by: where

is the electric field intensity,
is the force experienced by the test charge,
is the magnitude of the positive test charge placed in the field.
This definition implies that the electric field at a point is independent of the test charge used, provided the charge is small enough not to disturb the existing field.
S.I. Unit:
The S.I. unit of electric field intensity is derived from the units of force and charge. Since force is measured in newtons (N) and charge in coulombs (C), the unit of electric field intensity is: Alternatively, electric field can also be expressed in terms of potential difference and distance, giving the unit: Both units are equivalent, as .
Thus, the electric field quantifies how strongly and in which direction an electric force would act on a positive unit charge placed at any point in space.

According to Gauss's law, the net electric flux through any closed surface is directly proportional to the net charge enclosed within that surface. Mathematically, it is expressed as: where is the electric field, is the differential area vector on the closed surface, and is the permittivity of free space.
An electric dipole consists of two equal and opposite charges, and , separated by a small distance. Since these charges are equal in magnitude but opposite in sign, the total net charge enclosed by any closed surface that contains the dipole is: Therefore, by Gauss's law, the net electric flux through any closed surface enclosing an electric dipole is: This means that despite the presence of strong electric fields around the dipole, the total outward electric flux through a closed surface around it is zero because the positive and negative charges contribute equal and opposite fluxes that cancel each other out.
Hence, the net electric flux through a closed surface enclosing an electric dipole is always zero.

A metal is a perfect conductor containing free electrons that can move freely in response to an electric field. Due to this behavior, the dielectric constant (or relative permittivity) of a metal is effectively infinite: This implies that the metal can completely screen or cancel the electric field inside it, preventing any penetration of the field within the conductor.

Electric dipole moment is given by: where is the magnitude of one charge, and is the separation between the charges. Given: So,

For a charged metallic sphere, all excess charge resides on the outer surface of the conductor. This is because in electrostatic equilibrium, the free electrons within a conductor redistribute themselves in such a way that the electric field inside the conductor is zero. According to Gauss’s Law, the electric field inside a conductor is: This result follows from the fact that the net electric flux through a Gaussian surface entirely contained within the conductor is zero, implying that the electric field must also be zero. Hence, as we move from the surface to the center of the sphere, the electric field drops to zero instantly and remains zero throughout the interior of the sphere.

According to Gauss’s Law, the total electric flux through a closed surface depends only on the net charge enclosed: An electric dipole consists of two equal and opposite charges. The net charge of a dipole is zero. So, even if there are dipoles inside the surface, the total net charge enclosed remains zero:

The electric potential due to an electric dipole at a point depends on the angle between the dipole axis and the position vector. In the broad-side-on position, the point lies on the equatorial line of the dipole (i.e., perpendicular to the dipole axis), so . The potential at a point on the equatorial line is given by: Since , this would seem to give , but this formula is valid for general angles. On the equatorial line (broad-side-on), the correct potential is:

When a charge is accelerated through a potential difference , the energy gained by the charge is given by the relation: For an electron, the charge is: Given the potential difference: Substituting the values into the energy formula: So, the energy gained by the electron is electron volts (eV), which is equivalent to the energy an electron acquires when accelerated through a potential difference of 5 volts.

The electric field is related to the negative gradient of electric potential. In one-dimensional form, this relationship is expressed as: This equation indicates that the electric field is equal to the negative rate of change of electric potential with respect to distance . The negative sign shows that the electric field points in the direction of decreasing electric potential. In other words, a positive test charge placed in the field will naturally move from a region of higher potential to a region of lower potential.

The potential energy between two electric dipoles separated by a distance varies inversely as the cube of the distance between them. Mathematically, this is expressed as: This relationship arises from the interaction between the dipole moments and the electric fields produced by each dipole. The derivation involves vector calculus and the superposition principle, but the key takeaway is the inverse cubic dependence. None of the given options explicitly show an inverse power relationship. However, the closest match is option (B) if it is interpreted as implying an inverse cube dependence. Therefore, based on the nature of dipole-dipole interactions, the correct or most appropriate choice is option (B).

Linear charge density, denoted by , is defined as the charge per unit length. This is a way to describe how charge is distributed along a one-dimensional object, such as a wire or a thin rod. where:

is the total electric charge distributed along the length of the object,
is the total length over which the charge is distributed,
represents the amount of charge per unit length.
To understand the dimensional properties of , we examine the units of its components. The SI unit of charge is the coulomb (C), and the SI unit of length is the meter (m). Therefore, the unit of linear charge density is: This unit tells us how many coulombs of charge exist per meter of length. For example, a linear charge density of means there are 5 coulombs of charge distributed uniformly along each meter of the object.

The intensity of an electric field is proportional to the square of the electric field. The electric field intensity is defined as the force experienced per unit positive charge: Now, to find the dimensional formula of , we need to substitute the dimensional formulas of force and charge . The dimensional formula of force is: The dimensional formula of electric charge is: Substituting these into the formula for : Therefore, the dimensional formula of the intensity of the electric field is:

The charge of a single electron is . To find the number of electrons in a charge , we use the formula: Thus, the number of electrons present in 8 coulomb of negative charge is .

Step 1: Let the two charges be at points and , separated by distance . The point is at distance from the center .
Step 2: The distances from to the charges are: assuming .
Step 3: The potential at is:
Step 4: Using binomial approximation for :
Step 5: Substitute to get:
Step 6: Since , the expression becomes:

Step 1: Coulomb's law.
Coulomb’s law for the force between two charges and separated by a distance is: where is Coulomb’s constant.
Step 2: Applying changes.
When the distance is halved, . So the force increases by a factor of (since ). Also, when one charge is halved, , so the force decreases by a factor of .
Step 3: Final force.
Thus, the final force will be: Conclusion: The force will become double the original force.

The work done in rotating an electric dipole in a uniform electric field can be calculated using the formula: Where: - is the dipole moment,
- is the electric field,
- is the initial angle,
- is the final angle.
In this case, the dipole is initially aligned parallel to the electric field, so , and after rotating it through , . The dipole moment and electric field are parallel in the initial position, and the final position is antiparallel.
Substituting into the formula: Thus, the work done in rotating the dipole through an angle of is:

Step 1: Energy of the electron.
The kinetic energy gained by the electron when accelerated through a potential difference is given by: where is the charge of the electron.
Step 2: Relationship with speed.
The kinetic energy of the electron is also given by: where is the mass of the electron and is its velocity.
Step 3: Equating the two expressions.
Equating the kinetic energy expressions, we get: Solving for , we get: Thus, the speed is proportional to .
Step 4: Elimination.
- (A) proportional to : Incorrect, the speed is not directly proportional to .
- (B) proportional to : Correct, as derived above.
- (C) proportional to : Incorrect.
- (D) proportional to : Incorrect.
Step 5: Conclusion.
Hence, the speed of the electron is proportional to .

Step 1: Coulomb's Law.
The repulsive force between two charges is given by Coulomb’s law: where is the force, and are the magnitudes of the two charges, and is the distance between them. The constant is the permittivity of free space.
Step 2: Effect of dielectric material.
When a dielectric material is inserted between two positive charges, the permittivity of the medium increases, and as a result, the electrostatic force between the charges decreases. The dielectric constant reduces the effective force.
Step 3: Explanation of each option.
- (A) Increase: Incorrect. The force will not increase; it decreases due to the dielectric.
- (B) Decrease: Correct. The dielectric reduces the force between the charges.
- (C) Remain same: Incorrect. The presence of a dielectric changes the electrostatic force.
- (D) Become zero: Incorrect. The force does not become zero; it decreases.
Step 4: Conclusion.
Inserting a dielectric material decreases the repulsive force between the charges.

Step 1: Understanding capacitance.
The capacity (or capacitance) of a condenser depends on the geometry and separation of the plates. Specifically, capacitance is directly proportional to the area of the plates and inversely proportional to the distance between them.
Step 2: Impact of charge.
While the charge on the plates affects the energy stored in the condenser, it does not affect its capacitance. The capacitance is determined by the properties of the material and geometry, not the charge.
Step 3: Elimination.
- (A) shape of plates: The shape of the plates does affect the capacitance.
- (B) size of plates: Larger plates increase capacitance.
- (C) charge on plates: Correct, the capacitance does not depend on the charge.
- (D) distance between plates: Increasing the distance reduces the capacitance.
Step 4: Conclusion.
Thus, the capacity of a condenser does not depend on the charge on the plates.

Step 1: Electric field of a conductor.
For a charged conductor, the electric field just outside the surface is perpendicular to the surface. This is a result of Gauss's law, which states that the electric field at the surface of a conductor must be normal (perpendicular) to the surface in electrostatic equilibrium.
Step 2: Elimination.
- (A) parallel to the surface: Incorrect, as the field is not parallel to the surface.
- (B) perpendicular to the surface: Correct, the electric field is always perpendicular to the surface of a conductor.
- (C) at 45° angle to the surface: Incorrect, not the case for conductors.
- (D) zero: Incorrect, there is always an electric field on the surface of a charged conductor.
Step 3: Conclusion.
Thus, the electric field on the outer surface of a charged conductor is perpendicular to the surface.

Step 1: Behavior of electric field inside a conductor.
When a conductor is placed in an external electric field, free charges inside the conductor move in such a way that they cancel out the external field within the conductor. This results in the electric field inside the conductor becoming zero.
Step 2: Explanation of each option.
- (A) zero: Correct. Inside a conductor, the electric field is always zero in electrostatic equilibrium.
- (B) equal to the external field: Incorrect. The internal field is canceled out by the movement of free charges.
- (C) twice the external field: Incorrect. This does not hold for conductors.
- (D) half the external field: Incorrect. The field inside the conductor is zero, not a fraction of the external field.
Step 3: Conclusion.
The electric field inside the conductor is zero.

Step 1: Electric potential energy between charges.
The potential energy between two point charges is given by the formula: where is the Coulomb constant, and are the charges, and is the distance between them.
Step 2: Analyzing the sign of potential energy.
- When two point charges have opposite signs (one positive and one negative), their potential energy is always negative because the product of will be negative. This indicates an attractive interaction between the charges.
Step 3: Explanation of each option.
- (A) The potential energy is always negative: Correct. For opposite charges, the potential energy is negative due to the attractive force.
- (B) The potential energy is always positive: Incorrect. This happens only for like charges.
- (C) The potential energy can be either positive or negative: Incorrect for opposite charges, where the energy is always negative.
- (D) The potential energy is zero: Incorrect. The potential energy is not zero for opposite charges unless the charges are at infinite distance.
Step 4: Conclusion.
For two point charges of opposite signs, the potential energy is always negative.

Step 1: Gauss’s Law.
Gauss’s law states that the total electric flux through a closed surface is directly proportional to the total charge enclosed within the surface. It is mathematically expressed as: where is the electric flux, is the charge enclosed, and is the permittivity of free space.
Step 2: Analyze options.
- (A) Proportional to the charge enclosed: Correct. Gauss's law directly relates the electric flux to the enclosed charge.
- (B) Inversely proportional to the charge enclosed: Incorrect. The flux increases with the charge enclosed, not inversely.
- (C) Zero: Incorrect. The flux is not necessarily zero unless the charge enclosed is zero.
- (D) Proportional to the square of the charge enclosed: Incorrect. There is no square relation in Gauss’s law; the flux is directly proportional to the charge.
Step 3: Conclusion.
Gauss's law indicates that the electric flux is proportional to the charge enclosed.

The volume density of charge (also known as charge density) is the amount of electric charge per unit volume at a specific point in space. It is denoted by and is a measure of how much charge is present in a given volume of space. The formula for the volume density of charge is: Where: - is the total charge in the volume,
- is the volume in which the charge is distributed.
The SI unit of charge density is coulombs per cubic meter ( ). It expresses the amount of charge (in coulombs) per unit volume (in cubic meters).
In summary, the volume density of charge is the charge per unit volume, and its SI unit is .

Gauss's Theorem states that the total electric flux ( ) passing through any closed surface is proportional to the total charge enclosed within the surface. Mathematically, it is expressed as: Where: - is the surface integral of the electric field over the closed surface ,
- is the total charge enclosed within the surface,
- is the permittivity of free space.
Proof:
The flux through a closed surface is the integral of the electric field over the surface area. If the charge distribution is symmetric, such as in the case of spherical symmetry, the electric field is normal to the surface at all points, and the magnitude is constant. Thus, Gauss's law simplifies to: Where is the magnitude of the electric field and is the surface area. Using Gauss's law, we can calculate the electric field at any point outside a spherical charge distribution.
Electric Field Outside a Hollow Uniformly Charged Sphere:
For a uniformly charged spherical shell, we apply Gauss's theorem to find the electric field outside the sphere. Consider a spherical Gaussian surface of radius greater than the radius of the sphere. Since the charge is uniformly distributed, by symmetry, the electric field at any point on the surface will be radially outward and have the same magnitude at all points.
By Gauss's law, we have: Where is the surface area of the spherical Gaussian surface and is the total charge on the sphere. Therefore, the electric field intensity is given by: Thus, the electric field outside the hollow uniformly charged sphere behaves exactly as if all the charge were concentrated at the center of the sphere.

Step 1: Formula for Electric Potential Energy.
The formula for the electric potential energy of two point charges is given by: where is Coulomb's constant, , and .
Step 2: Substituting the values.
Conclusion: The electric potential energy of the system is joule.

Step 1: Definition of capacitance.
Capacitance is the ability of a system to store charge per unit potential difference. The unit of capacitance is the farad (F), which is defined as: Step 2: Understanding the units.
The unit of capacitance is , which is choice (D).
Step 3: Elimination of options.
- (A) coulomb: Incorrect, unit of charge, not capacitance.
- (B) ampere: Incorrect, unit of current.
- (C) volt: Incorrect, unit of potential difference.
- (D) coulomb/volt: Correct, unit of capacitance.
Conclusion: The unit of capacitance is coulomb/volt.